Understanding geometrical phases in quantum mechanics: An elementary example

Joy Christian

Yu-xi Liu

We present a geometrical way of understanding the dynamics of wavefunctions in a free space, using the phase-space formulation of quantum mechanics. By visualizing the Wigner function, the spreading, shearing, the so-called "negative probability flow" of wavefunctions, and the long-time asymptotic dispersion, are intuited visually. These results are not new, but previous derivations were analytical, whereas this paper presents elementary geometric arguments that are almost "proofs without words", and suitable for a first course in quantum mechanics.

Tommy Chan Sze-wing

H. J. Korsch, B Schellhaaß

We discuss the basic properties of momentum distributions in quantum mechanics for elementary systems as well as their classical analogue. Semiclassical approximations can show a quantitative connection between the classical and quantum cases. We believe that such distributions provide a useful tool to improve the understanding of elementary quantum mechanics. Important differences between distributions in coordinate and momentum space are pointed out. Elementary examples (free and uniformly accelerating particle, harmonic oscillator and square-well potential) are discussed.

P. Lebœuf, Amaury Mouchet

Degeneracies in the spectrum of an adiabatically transported quantum system are important to determine the geometrical phase factor, and may be interpreted as magnetic monopoles. We investigate the mechanism by which constraints acting on the system, related to local symmetries, can create arbitrarily large monopole charges. These charges are associated with different geometries of the degeneracy. An explicit method to compute the charge as well as several illustrative examples are given.

Arno Böhm